GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 2, 1995, 151-164

SEQUENTIAL CONVERGENCE IN TOPOLOGICAL VECTOR SPACES

A. K. KATSARAS AND V. BENEKAS

Abstract. For a given linear topology τ, on a vector space E, the finest linear topology having the same τ convergent sequences, and the finest linear topology on E having the same τ precompact sets, are investigated. Also, the sequentially bornological spaces and the sequentially barreled spaces are introduced and some of their properties are studied.

Introduction

For a locally convex space E, Webb constructed in [1] the finest locally convex topology on E having the same convergent sequences as the initial topology and the finest locally convex topology having the same precompact sets as the initial topology.

In this paper, using the notion of a string introduced in [2], we construct the finest linear topology which has the same convergent sequences as the initial linear topology and the finest linear topology having the same precompact sets. Some of the properties of these topologies are studied. We also give the notion of a sequentially bornological space. A space E is Sbornological if every bounded linear map from E to an arbitrary topological vector space is sequentially continuous. This is equivalent to the following: Every bounded pseudo-seminorm on E is sequentially continuous. The concept of a sequentially barreled space is also given and some of the properties of the S-bornological spaces and the S-barreled spaces are investigated.

1. Preliminaries

All vector spaces considered in this paper will be over the field of real numbers or the field of complex numbers. Following [2], we will call a string, in a vector space E, any sequence U = (V_{n}) of balanced absorbing sets such

151 that V_{n+1} + V_{n+1} ⊂ V_{n} for all n. A string U = (V_{n}), in a topological vector space E, is called topological if every V_{n} is a neighborhood of zero. The string U is called closed (resp. bornivorous) if every V_{n} is closed (resp. bornivorous). Every string U = (V_{n}) defines a linear topology τ + U having as a base at zero the sequence {V_{n} : n ∈ N}. The set N_{U} ∩_{n} V_{n} is a subspace of E. We will denote by E_{U} the quotient space E/N_{U}. The space E_{U} with the quotient topology is a metrizable topological vector space. By Ê_{U} we will denote the completion of E_{U}, and so Ê_{U} is a Frechet space and is a complete metrizable topological vector space. Any family F of strings in E defines a linear topology, namely the supremum of the topologies τ_{U}, U ∈ F. A family F of topological strings, in a topological vector space E, is called a base for the topological strings if for any topological string W = (W_{n}) there exists U = (V_{n}) in F with U ≤ W, where U ≤ W means that V_{n} ⊂ W_{n} for all n. The family of all closed topological strings is a base for the topological strings.

1991 Mathematics Subject Classification. 46 A.

Key words and phrases. String, sequential space, sequentially barreled space, sequentially bornological space, sequential convergence.

1072-947X/95/0300-0151$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

Definition 1.1 ([3,6.1]). A pseudo-seminorm, on a vector space E over 𝕂, is a function p : E → ℝ with the following properties:

1) p(x + y) ≤ p(x) + p(y), for all x, y in E.

2) p(λx) ≤ p(x) for all x ∈ E when |λ| ≤ 1.

3) If λ_{n} → 0, then 𝕂 for all p(λ_{n}x) → 0 in x ∈ E.

4) If p(x_{n}) → 0, then p(λ_{n}x) → 0 for all λ in 𝕂. If p(x) > 0 when x ≠ 0, then p is called a pseudo-norm.

Lemma 1.2. If p is any pseudo-seminorm on E, then d_{p} : E × E → ℝ, d_{p}(x, y) = p(x − y), is a pseudo-metric, and the corresponding topology τ_{p} is linear. If p is a pseudo-norm, then d_{p} is a metric.

Lemma 1.3. If p is a pseudo-seminorm on E and V_{n} = {x : p(x) ≤ 2^{−n}}, then U = (V_{n}) is a string and τ_{U} = τ_{p}.

Lemma 1.4 ([3,6.1]). If U = (V_{n}) is a string in E, then there exists a pseudo-seminorm p_{U} on E such that

{x ∈ E : p_{U} (x) ≤ 2^{−n−1}} ⊂ V_{n} ⊂ {x : p_{U} (x) ≤ 2^{−n}}

and so τ_{U} = τ_{p_{U}}.

Any linear topology on E is generated by the family of all continuous pseudo-seminorms on E.

Lemma 1.5. Let (p_{n}) be a sequence of pseudo-seminorms on E and de-

fine p on E by p(x) = ∑_{n=1}^{∞} \frac{1}{2^{n}} · \frac{p_{n}(x)}{1 + p_{n}(x)}. Then p is a pseudo-seminorm

on E and τ_{p} coincides with the topology generated by the sequence (p_{n}) of pseudo-seminorms.

SEQUENTIAL CONVERGENCE IN TOPOLOGICAL VECTOR SPACES 153

Lemma 1.6. A topological vector space E is pseudo-metrizable iff its topology is generated by a countable family of pseudo-seminorms and this is true iff the topology of E is generated by a single pseudo-seminorm.

2. Sequential Spaces

Definition 2.1. Let V be a subset of a topological vector space E. Then V is called a sequential neighborhood (S-neighborhood) of zero if every null sequence in E lies eventually in V. The set V is called sequentially closed (S-closed) if every x in E, which is a limit of a sequence in V, belongs to V.

Definition 2.2. A string U = (V_{n}), in a topological vector space E, is called S-topological if every V_{n} is an S-neighborhood of zero.

Lemma 2.3. Let U = (V_{n}) be a string in a topological vector space E and let p be a pseudo-seminorm on E such that τ_{p} = τ_{U}. Then, U is Stopological iff p is sequentially continuous.

Proof. Let U be S-topological and let (x_{n}) be a sequence in E converging to some x. Given ε > 0, the set W = {x : p(x) ≤ ε} is a τ_{p}-neighborhood of zero and so there is m such that V_{m} ⊂ W. Since x_{n} − x → 0 in E and V_{m} is an S-neighborhood, there exists n_{0} such that x_{n} − x ∈ V_{m} when n ≥ n_{0}, and so p(x_{n} − x) ≤ ε if n ≥ n_{0}. The converse is proved analogously.

Definition 2.4. A topological vector space E is said to be a sequential space if every S-topological string in E is topological.

Proposition 2.5. Let U = (V_{n}) be a string in a topological vector space and let Ê_{U} be the completion of E_{U}. If J = J_{U} : E → E_{U} is the canonical mapping, then the following are equivalent:

(1) U is S-topological in E.

(2) J is sequentially continuous.

(3) J is sequentially continuous as a map from E to Ê_{U}.

Proof. It is clear that (1)⇒(2)⇒(3).

(3)⇒(1). For each subset A of E_{U}, let \overline{A} denote its closure in Ê_{U}. Then, the family \{\overline{J(V_{n})} : n ∈ N\} is a base at zero in Ê_{U}. Since the closure of J(V_{n+2}) in E_{U} is contained in J(V_{n+2}) + J(V_{n+2}) ⊂ J(V_{n+1}), we have

J^{−1}(\overline{V_{n+2})}) = J^{−1}(J(\overline{V_{n+2}}) ∩ E_{U}) ⊂ J^{−1}(J(V_{n+1})) =
= V_{n+1} + N_{U} ⊂ V_{n+1} + V_{n+1} ⊂ V_{N},

which shows that V_{n} is an S-neighborhood of zero in E since J is sequentially continuous.

Proposition 2.6. For a topological vector space E, the following are equivalent:

(1) E is sequential.

(2) Every sequentially continuous linear map, from E to an arbitrary topological vector space, is continuous.

(3) Every sequentially continuous linear map, from E to an arbitrary Frechet space, is continuous.

(4) Every sequentially continuous pseudo-seminorm on E is continuous.

Proof. (1)⇒(2). Let f : E → F be a linear and sequentially continuous and let V be a neighborhood of zero in F. There exists a topological string U = (V_{n}) in F with V_{1} = V. Since f is sequentially continuous, f^{−1}(U) is S-topological and hence topological by (1). Hence f^{−1}(V) is a neighborhood of zero in E.

(3)⇒(1). Let U = (W_{n}) be an S-topological string. By Proposition 2.5, the canonical mapping J : E → Ê_{U} is sequentially continuous and hence by (3) continuous. As in the proof of the implication (3)⇒(1) in the preceding Proposition, it follows that each V_{n} is a neighborhood of zero in E.

(1)⇒(4). Let p be a sequentially continuous pseudo-seminorm and set V_{n} = {x : p(x) ≤ 2^{−n}}. Then U = (V_{n}) is an S-topological string and hence topological, which implies that p is continuous.

(4)⇒(1). Let U = (W_{n}) be an S-topological string and let S be as in Lemma 1.4. By Lemma 2.3, p_{U} is sequentially continuous and hence by (4) continuous, which implies that (W_{n}) is topological.

Proposition 2.7. Let {E_{α} : α ∈ I} be a family of topological vector spaces, E a vector space and, for each α, f_{α} : E_{α} → E a linear map. If τ is the finest linear topology on E for which each f_{α} is continuous and if each E_{α} is sequentially, then (E, τ) is sequential.

Proof. If U is an S-topological string in (E, τ), then each f_{α}^{−1}(U) is Stopological in E_{α} and so it is topological in E, which implies that U is topological in (E, τ).

Corollary 2.8. Quotient space and direct sums of sequential spaces are sequential.

The family F_{S} of all S-topological strings in (E, τ) is direct. We will denote by τ^{S} the linear topology generated by the family F_{S}.

Proposition 2.9. (1) τ^{S} is the coarsest sequential topology finer than τ.

(2) τ^{S} is the finest linear topology on E having the same with τ convergent sequences.

(3) The topologies τ and τ^{S} have the same bounded sets.

(4) If τ_{1} is a linear topology such that every τ-null sequence is also τ_{1}-null, then τ_{1} ≤ τ^{S}.

(5) A linear map f, from (E, τ) to an arbitrary topological vector space F, is sequentially continuous iff it is τ^{S}-continuous.

(6) τ^{S} is generated by the family of all sequentially continuous pseudoseminorms on (E, τ).

Proof. Clearly τ ≤ τ^{S}. It follows directly from the definition that every τ^{S}-null sequence is also τ^{S}-null and so τ and τ^{S} have the same convergent sequences and the same bounded sets. If τ_{1} is a linear topology on E such that every τ-null sequence is also τ_{1}-null, then every τ_{1}-topological string is sequentially τ-topological, which implies that τ_{1} ≤ τ^{S}. This proves (2),(3), and (4).

(1) Since every τ^{S}-sequential string is also τ-sequential (by (2)), the topology τ^{S} is sequential and it is clearly the coarser sequential topology finer than τ.

(5) If the linear map f : (E, τ) → F is sequentially continuous, then for each topological string U in F, f^{−1}(U) is an S-topological string in (E, τ) and so f^{−1}(U) is τ^{S}-topological, which proves that f is τ^{S}-continuous. The converse is clear.

(6) It follows from Lemma 2.3.

Proposition 2.10. If the linear map f : (E, τ) → (F, τ_{1}) is continuous, then f is (τ^{S}, τ_{1}^{S})-continuous.

Proof. It follows from the fact that, for each S-topological string in F, f^{−1}(U) is S-topological in E.

Corollary 2.11. If (E, τ) = ∏_{α∈I}(E_{α}, τ_{α}), then τ^{S} ≥ ∏_{α∈I} τ_{α}^{S}.

Proposition 2.12. If (E, τ) = ∏_{k=1}^{n}(E_{k}, τ_{n}), then τ^{S} = ∏_{k=1}^{n} τ_{k}^{S}.

Proof. Let τ_{0} = ∏_{k=1}^{n} τ_{k}^{S}. By the preceding corollary, we have τ_{0} ≤ τ^{S}.

On the other hand, let U(V_{n}) be an S-topological string in (E, τ). For each k, 1 ≤ k ≤ n, let j_{k} : E_{k} → E be the canonical mapping. Set V_{k}^{m} = j_{k}^{−1}(V_{m+n+1}). Each V_{k}^{m} is an S-neighborhood of zero in (E_{k}, τ_{k}) and so U_{k} = (V_{k}^{m})_{m=1}^{∞} is an S-topological string in E_{k}. The set

W_{m} = ∏_{k=1}^{n} V_{k}^{m}

is a τ_{0}-neighborhood of zero in E and W_{m} ⊂ V_{m}, which implies that U is τ_{0}-topological. Thus τ^{S} ≤ τ_{0}, and the result follows.

Proposition 2.13. Let (E, τ) = ⊕_{α∈I}(E_{α}, τ_{α}), where each (E_{α}, τ_{α}) is

Hausdorff. Then τ^{S} = ⊕_{α∈I}τ_{α}^{S}.

Proof. Let σ = ⊕_{α∈I}τ_{α}^{S}. If f_{α} : E_{α} → E is the canonical mapping, then f_{α}

is (τ_{α}^{S}, τ^{S})-continuous and so τ^{S} ≤ σ. On the other hand, let U = (V_{n}) be an S-topological string in (E, τ) and let (x^{(n)}) be a null sequence in (E, τ). By [2, 5.10], there exists a finite subset I_{0} = {α_{1}, …, α_{m}} of I such that x_{α}^{(n)} = 0 for all n and all α ∉ I_{0}. Let

g : (E, τ) → ⊕^{m}_{k=1}(E_{α_{k}}, τ_{α_{k}}) = ∏_{k=1}^{m} (E_{α_{k}}, τ_{α_{k}})

be a canonical projection. If τ_{0} = ∏_{k=1}^{m} τ_{α_{k}}, then g is (τ, τ_{0})-continuous and

so g is (τ^{S}, τ_{0}^{S})-continuous. The canonical embedding

h : ⊕^{m}_{k=1}(E_{α_{k}}, τ_{α_{k}}^{S}) → (E, σ)

is continuous. But

⊕^{n}_{k=1}τ_{α_{k}}^{S} = ∏_{k=1}^{n} τ_{α_{k}}^{S} = τ_{0}^{S}.

Thus, h is (τ_{0}^{S}, σ)-continuous and so

h ∘ g : (E, τ^{S}) → (E, σ)

is continuous and hence x^{(n)} = h ∘ g(x^{(n)}) → 0 in (E, σ). This proves that the identity map from (E, τ^{S}) to (E, σ) is sequentially continuous and therefore continuous. This implies that σ ≤ τ^{S} and the result follows.

Proposition 2.14. Let F be a topologically complemented subspace, of a topological vector space (E, τ), and let τ|F be the topology induced by τ on F. Then τ^{S}|F = (τ|F)^{S}.

Proof. Since the inclusion map from F to E is ((τ|F)^{S}, τ^{S})-continuous, we have that τ^{S}|F ≤ (τ|F)^{S}. To prove the inverse inequality, let G be a topological complement of E and let π_{1}, π_{2} be the projections of E onto F and G, respectively. Then π_{1}, π_{2} are continuous. Let now τ_{1} = τ|F and let U = (V_{n}) be an S-topological string in (F, τ_{1}). Let W_{n} = V_{n} + G. Each W_{n} is an S-neighborhood of zero in (E, τ). In fact, let (x_{n}) be a τ-null sequence. Since π_{1}(x_{m}) → 0 in F, there exists m_{0} such that π_{1}(x_{m}) ∈ V_{n} if m ≥ m_{0} and so x_{0} ∈ W_{n} if m ≥ n_{0}. Now, W = (W_{n}) is an S-topological string in (E, τ). Since W_{n} ∩F = V_{n}, it follows that U is (τ^{S}|F)-topological, which proves that τ_{1}^{S} ≤ τ^{S}|F.

3. The Topology τ^{p}

Definition 3.1. A string U = (V_{n}) in a topological vector space (E, τ) is called precompactly-topological (pr-topological) if for each n and each τ-precompact subset A of E there exists a finite subset S_{n} of A such that A ⊂ S_{n} + V_{n}.

Lemma 3.2. If U = (V_{n}) and W = (W_{n}) are pr-topological strings in E and λ a non-zero scalar, then λU = (λV_{n}) and U ∩ W = (V_{n} ∩ W_{n}) are also pr-topological.

Proof. It is easy to see that λU is pr-topological. For the U ∩ W, let A be a non-empty τ-precompact set and let k ∈ N. There are x_{1}, …, x_{n}, y_{1}, …, y_{m} in A such that

A ⊂ ⋃_{i=1}^{n} (x_{i} + V_{k+1}), A ⊂ ⋃_{j=1}^{n} (y_{j} + W_{k+1}).

Let

D = {(i, j) : A ∩ (x_{i} + V_{k+1}) ∩ (y_{j} + W_{k+1}) ≠ ∅}.

Clearly,

A ⊂ ⋃_{(i,j)∈D} (x_{i} + V_{k+1}) ∩ (y_{j} + W_{k+1}).

For each α = (i, j) ∈ D, choose z_{α} ∈ A ∩ (x_{i} + V_{k+1}) ∩ (y_{j} + W_{k+1}). If z ∈ (x_{i} + V_{k+1}) ∩ (y_{j} + W_{k+1}), then

z − x_{i} ∈ V_{k+1}, z_{α} − x_{i} ∈ V_{k+1}, z − y_{j} ∈ W_{k+1}, z_{α} − y_{j} ∈ W_{k+1}.

Thus

z − z_{α} = (z − x_{i}) − (z_{α} − x_{i}) ∈ V_{k+1} + V_{k+1} ⊂ V_{k}

and similarly z − z_{α} ∈ W_{k}, i.e., z − z_{α} ∈ V_{k} ∩ W_{k}. Thus A ⊂ {z_{α} : α ∈ D} + W_{k} ∩ V_{k}.

Notation 3.3. For a linear topology τ on E, we will denote by τ^{p} the linear topology generated by the family of all pr-topological strings in (E, τ).

We omit the proof of the following easily established proposition.

Proposition 3.4. 1) τ^{p} is the finest linear topology on E having the same τ precompact sets.

2) τ ≤ τ^{p} and if τ_{1} is a linear topology on E such that every τ-precompact set is also τ_{1}-precompact, then τ_{1} ≤ τ^{p}.

Proposition 3.5. A linear map f : (E, τ) → F is τ^{p}-continuous iff it maps τ-precompact sets in E into precompact sets in F.

Proof. The condition is clearly necessary since every τ-precompact set is also τ^{p}-precompact and images of precompact sets, under continuous linear mappings, are precompact. For the sufficiency, let U = (V_{n}) be a topological string in F. If A is a τ-precompact set, then f(A) is precompact in F and so, given n, there exists a finite subset S_{n} of A such that f(A) ⊂ f(S_{n}) + V_{n}, which implies that A ⊂ S_{n} + f^{−1}(V_{n}). Thus f^{−1}(U) is pr-topological in (E, τ), which proves that f is τ^{p}-continuous.

Proposition 3.6. If the linear map f : (E, τ) → (F, τ_{1}) is continuous, then f is (τ^{p}, τ_{1}^{p})-continuous.

Proof. It follows from the fact that for every pr-topological string in (F, τ_{1}), f^{−1}(U) is pr-topological in (E, τ).

Corollary 3.7. If (E, τ) = ∏_{α∈I}(E_{α}, τ_{α}), then τ^{p} ≥ ∏_{α∈I} τ_{α}^{p}.

For the proof of the following Proposition we use an argument analogous to that of Proposition 2.12.

Proposition 3.8. If (E, τ) = ∏_{k=1}^{n}(E_{k}, τ_{k}), then τ^{p} = ∏_{k=1}^{n} τ_{k}^{p}.

Proposition 3.9. Let (E, τ) = ⊕_{α∈I}(E_{α}, τ_{α}), where each τ_{α} is Hausdorff.

Then τ^{p} = ⊕_{α∈I}τ_{α}^{p}.

Proof. Let σ = ⊕_{α∈I}τ_{α}^{p}. If f_{α} : E_{α} → E is the canonical embedding, then f_{α}

is (τ_{α}^{p}, τ^{p})-continuous and so τ^{p} ≤ σ. On the other hand, let U = (V_{n}) be a σ-topological string. We need to show that U is pr-topological in (E, τ). To this end, let A be a τ-precompact set. By [2, 5.10], there exists a finite subset I′ = {α_{1}, …, α_{n}} of I such that x_{α} = 0 for all x ∈ A and all α ∉ I′. If π_{k} : E → E_{α_{k}} is the canonical projection, then A_{k} = π_{k}(A) is precompact in E_{α_{k}}. Given n, there exists a finite subset S of A such that

A_{k} ⊂ π_{k}(S) + f_{α_{k}}^{−1}(V_{n+m−1})

for k = 1, …, m. Given x ∈ A, there are x_{k} ∈ A_{k}, k = 1, …, m, such that

x = ∑_{k=1}^{m} f_{α_{k}}(x_{k}). Since

\underbrace{V_{n+m−1} + V_{n+m−1} + ⋯ + V_{n+m−1}}_{m−times} ⊂ V_{n}

and since f_{α_{k}}(x_{k}) ∈ f_{α}(S_{k}) + V_{m+n−1}, S_{k} = π_{k}(S), it follows that x ∈

T + V_{n}, T = ∑_{k=1}^{m} f_{α_{k}}(S_{k}). Thus, A ⊂ T + V_{n}, which shows that U is

pr-topological in (E, τ). This clearly completes the proof.

Proposition 3.10. Let F be a topologically complemented subspace of a topological vector space (E, τ). Then τ^{p}|F = (τ|F)^{p}.

Proof. For the proof we use an argument analogous to that of Proposition 2.14.

Proposition 3.11. Let {(E_{α}, τ_{α}) : α ∈ I} be a family of topological vector spaces, E a vector space, and f_{α} : E_{α} → E a linear map, for each α ∈ I. If τ is the finest linear topology on E for which each f_{α} is continuous and if τ_{α}^{p} = τ^{p} for all α ∈ I, then τ^{p} = τ.

Proof. If U is a pr-topological string in (E, τ), then f_{α}^{−1}(U) is pr-topological in (E_{α}, τ_{α}) and so f^{−1}(U) is τ_{α}-topological, which implies that U is τtopological.

Corollary 3.12. Let F be a subspace of (E, τ) and let τ_{0} be the quotient topology on E|F. If τ^{p} = τ, then τ_{0}^{p} = τ_{0}.

Lemma 3.13. Let τ_{1}, τ_{2} be two linear topologies on E. If every τ_{1}- precompact set is also τ_{2}-precompact, then every τ_{1}- bounded set is τ_{2}-bounded.

Proof. Assume the set A is τ_{1}-bounded but not τ_{2}-bounded. Then there exists a balanced τ_{2}-neighborhood V of zero in E and a sequence (x_{n}) in A such that x_{n} ∉ n^{2}V. If y_{n} = n^{−1}x_{n}, then y_{n} →^{τ_{1}} 0 (since A is τ_{1}-bounded), and so the set B = {y_{n} : n ∈ N} is τ_{1}-precompact. By our hypothesis, B is τ_{2}-precompact and hence B is τ_{2}-bounded, which is not true since B is not absorbed by V.

Corollary 3.14. 1) τ and τ^{p} have the same bounded sets.

2) If (E, τ) is a bornological space, then τ = τ^{S} = τ^{p}.

Recall that a topological vector space E is called countably barreled if every string in E which is the intersection of a countable number of closed topological strings is topological. Also, E is called countably quasi-barreled if every bornivorous string in E which is the intersection of a countable number of closed topological strings is topological.

Notation 3.15. If τ is a linear topology on E, we will denote by τ^{β} the linear topology generated by the family of all τ-closed strings. By τ^{β∗} we will denote the linear topology generated by the family of all closed bornivorous strings in (E, τ).

Proposition 3.16. Let (E, τ) be a topological vector space.

1) If (E, τ) is countably quasi-barreled, then τ^{β∗} ≤ τ^{p}.

2) If (E, τ) is countably barreled, then τ^{β} ≤ τ^{p}.

Proof. It follows from [2, 18.5] since every τ-precompact set is τ^{β∗} (resp. τ^{β})-precompact when (E, τ) is countably quasi-barreled (resp. countably barreled).

4. Sequentially Bornological Spaces

Definition 4.1. A topological vector space E is called sequentially bornological (S-bornological) if every bornivorous string is S-topological.

Since τ and τ^{S} have the same bounded sets, (E, τ) is S-bornological iff (E, τ^{S}) is bornological.

Definition 4.2. A pseudo-seminorm p, on a topological vector space E, is called bounded if for every bounded sequence (x_{n}) in E and every null sequence of scalars (λ_{n}) we have p(λ_{n}x_{n}) → 0.

Note. If p is a seminorm, then p is bounded iff it is bounded on bounded sets.

Definition 4.3. A linear map f between two topological vector spaces E, F is called bounded if it maps bounded sets in E into bounded sets in F.

Proposition 4.4. For a topological vector space (E, τ) the following properties are equivalent:

(1) (E, τ) is S-bornological.

(2) Every bounded linear map, from (E, τ) to an arbitrary topological vector space, is sequentially continuous.

(3) Every bounded linear map, from (E, τ) to an arbitrary Frechet space, is sequentially continuous.

(4) Every bounded pseudo-seminorm on E is sequentially continuous.

Proof. (1)⇒(2). Let f : (E, τ) → F be linear and bounded. If U is a topological string in F, then f^{−1}(U) is a bornivorous string in (E, τ), and hence f^{−1}(U) is S-topological. This proves that f is τ^{S}-continuous and hence f is sequentially τ-continuous.

(3)⇒(1). Let U = (V_{n}) be a bornivorous string in E and consider the Frechet space F = Ê_{U}. Let j : E → F be the canonical mapping. If A is bounded in (E, τ), then for each n there exists a scalar λ such that A ⊂ λV_{n} and so j(A) ⊂ λj(V_{n}), which proves that J(A) is bounded in F. Thus, j is bounded and hence j is sequentially continuous by our hypothesis, which implies that U is S-topological by Proposition 2.5.

(1)⇒(4). Let p be a bounded pseudo-seminorm on (E, τ) and set W_{m} = {x : p(x) ≤ 2^{−m}}. Every W_{m} is bornivorous. In fact, assume that W_{m} does not absorb a τ-bounded set A. Then, there exists a sequence (x_{n}) in A with x_{n} ∉ 2^{n}W_{m}. Now (x_{n}) is a bounded sequence and p(2^{−n}x_{n}) > 2^{−m} for all n, which is a contradiction. Thus, U = (W_{n}) is a bornivorous string and hence S-topological by (1). Let now m be given and let (x_{n}) be a τ-null sequence. Since W_{m} is an S-neighborhood of zero, there exists n_{0} such that x_{n} ∈ W_{m} and so p(x_{n}) ≤ 2^{−} if n ≥ n_{0}.

(4)⇒(1). Let W = (V_{n}) be a bornivorous string in (E, τ) and let p = p_{W} be as in Lemma 1.4. Then, p is bounded. In fact, let (x_{n}) be a bounded sequence and let λ_{n} → 0. Given k, there exists a scalar α such that (x_{n}) ⊂ αV_{k}. Now

λ_{n}x_{n} ∈ λ_{n} αV_{k} ⊂ V_{k}

for large n, which implies that p(λ_{n}x_{n}) ≤ 2^{−k} eventually. By (4), p is sequentially continuous and hence W is S-topological by Lemma 2.3.

Proposition 4.5. A countable product (E, τ) = ∏_{n=1}^{∞}(E_{k}, τ_{k}) of S-borno-

logical spaces is S-bornological.

Proof. Each τ_{n}^{S} is bornological and so τ_{0} = ∏_{n=1}^{∞} τ_{n}^{S} is bornological by [2, 11.6].

If τ^{b} is the bornological topology associated with τ, then τ^{S} ≤ τ^{b}, since τ^{S} has the same τ bounded sets. Since τ ≤ τ_{0} ≤ τ_{S} (by Corollary 2.11), we have that τ_{0} = τ^{b} = τ^{S} and so τ^{S} is bornological.

Proposition 4.6. Let F be a topologically complemented subspace of a topological vector space (E, τ). If (E, τ) is S-bornological, then F is Sbornological.

Proof. Let τ_{0} = τ|F. By Proposition 2.14, we have τ_{0}^{S} = τ^{S}|F. Thus, it suffices to assume that (E, τ) is bornological and show that τ_{0} is bornological. If U = (V_{n}) is a bornivorous string in (F, τ_{0}) and if G is a topological complement of F, then (W_{n}), where W_{n} = V_{n} + G is a bornivorous string in (E, τ) and W_{n} ∩ F = V_{n}, which shows that U is τ_{0}-topological.

Proposition 4.7. Let (E, τ) = ⊕_{α∈I}(E_{α}, τ_{α}), where each τ_{α} is Hausdorff.

If each (E_{α}, τ_{α}) is S-bornological, then (E, τ) is S-bornological.

Proof. By Proposition 2.13, we have τ^{S} = ⊕_{α∈I}τ_{α}^{S}. Since the direct sum of

bornological spaces is bornological ([2, 11.5]), the result follows.

5. Sequentially Barreled Spaces

Definition 5.1. A topological vector space E is called sequentially-bar-reled (S-barreled) if every closed string in E is S-topological.

Proposition 5.2. A topological vector space E is S-barreled iff every lower-semicontinuous pseudo-seminorm on E is sequentially continuous.

Proof. Assume that E is S-barreled and let p be a lower-semicontinuous pseudo-seminorm on E. If V_{n} = {x : p(x) ≤ 2^{−n}}, then U = (V_{n}) is a closed string and hence S-topological, which implies that p is sequentially continuous (by Lemma 2.3). Conversely, suppose that every lower-semicontinuous pseudo-seminorm on E is sequentially continuous and let W = (W_{n}) be a closed string in E. Let p = p_{W} be as in Lemma 1.4. Let {V_{γ} : γ ∈ Γ} be a base at zero in E consisting of balanced sets and make Γ into a direct set by defining γ_{1} ≥ γ_{2} if V_{γ_{1}} ⊂ V_{γ_{2}}. Define p on E by

p(x) = sup_{γ∈Γ} inf_{y∈V_{γ}} p_{W} (x + y).

Then, p is lower-semicontinuous (it is the biggest lower-semicontinuous function f with f ≤ p) and it is easy to show that p(x + y) ≤ p(x) + p(y) and that p(λx) ≤ p(x) when |λ| ≤ 1. Moreover

{x : p(x) < 2^{−n−1}} ⊂ W_{n}.  (∗)

In fact, let p(x) < 2^{−n−1}. For each γ ∈ Γ, there exists y_{γ} in V_{γ} with p_{W} (x + y_{γ}) < 2^{−n−1} and so x + y_{γ} ∈ W_{n}. Since the net (y_{γ})_{γ∈Γ} converges to zero and W_{n} is closed, we have that x ∈ W_{n}. It now follows easily that p is a pseudo-seminorm which, by our hypothesis, is sequentially continuous. Now (∗) implies that W is S-topological and the result follows.

Proposition 5.3. A topological vector space (E, τ) is S-barreled iff any linear topology τ_{1} on E, having a base at zero consisting of τ-closed sets, is weaker than τ^{S}.

Proof. Suppose that (E, τ) is S-barreled and let τ_{1} have the stated properties. Then, there is a base F for the τ_{1}-topological strings, consisting of τ-closed strings. Since (E, τ) is S-barreled, every member of F is Stopological in (E, τ) and hence τ_{1} ≤ τ^{S}. Conversely, assume that (E, τ) has the same property stated in the proposition. The family F_{c} of all τ-closed strings is directed and it generates a linear topology τ_{2}. This topology has a base at zero consisting of τ-closed sets and hence τ_{2} ≤ τ^{S} by our hypothesis. It follows that every τ-closed string is S-topological in (E, τ) and so (E, τ) is S-barreled.

Definition 5.4. A family F of linear maps, from a topological vector space E to another F, is called sequentially equicontinuous if for each neighborhood V of zero in F the set ∩_{f∈F}f^{−1}(V) is an S-neighborhood of zero

in E.

Proposition 5.5. For a topological vector space E, the following properties are equivalent:

(1) E is S-barreled.

(2) Every pointwise bounded family F of continuous linear maps, from E into another topological vector space F, is S-equicontinuous.

(3) Every pointwise bounded family of continuous linear maps, from E into an arbitrary Frechet space, is S-equicontinuous.

Proof. (1)⇒(2). Let V be a neighborhood of zero in F. Choose a closed topological string U = (V_{n}) in F with V_{1} ⊂ V. Let W_{n} = ∩_{f∈F}f^{−1}(V_{n}).

Then (W_{n}) is a closed string in E and hence S-topological by (1). Thus ∩_{f∈F}f^{−1}(V) is an S-neighborhood because it contains W_{1}.

(3)⇒(1). For the proof we use an argument analogous to the one used in [2, 7.2].

Corollary 5.6. Let E be an S-barreled space, F an arbitrary topological vector space, and (f_{α})_{α∈I} a pointwise bounded net of continuous linear maps from E to F. If the limit lim f_{α}(x) = f(x) exists for each x ∈ E, then f is sequentially continuous.

Corollary 5.7. Every pointwise limit of continuous linear maps, from an S-barreled space into an arbitrary topological vector space, is sequentially continuous.

Proposition 5.8. Let E be a Hausdorff S-barreled space and let F be a subspace of finite codimension. Then, F is S-barreled.

Proof. We may consider the case in which the codimension of F is one. Let U = (V_{n}) be a closed string in F. There are two possible cases.

Case I. Each V_{n} is a proper subset of its closure \overline{V}_{n} in E. Then, each \overline{V}_{n} is absorbing in \overline{E}. In fact, let x_{0} ∈ \overline{V}_{n+1}\V_{n+1} and let x ∈ E. Then x = y + λx_{0} with y ∈ F. Since V_{n+1} is absorbing in F, there exists a scalar μ, |μ| ≥ |λ|, with y ∈ μV_{n+1}, and so

x ∈ μV_{n+1} + λ\overline{V}_{n+1} ⊂ μ\overline{V}_{n}.

Now, (\overline{V}_{n}) is a closed string in (E, τ) and hence each \overline{V}_{n} is an S-neigh-borhood in (E, τ). Moreover, \overline{V}_{n} ∩ F = V_{n}. It is clear now that U is S-topological in F.

Case II. There exists an n such that V_{n} is closed in E. We will show that each V_{m} is an S-neighborhood in F. Without loss of generality, we may assume that V_{1} is closed in F. Then, each V_{m} is closed in E. Choose x_{0} ∈ E with x_{0} ∉ F and let W = {λx_{0} : |λ| ≤ 1}. The set W is compact in E and hence each W_{n} = V_{n} + 2^{−n}W is closed in E. Clearly (W_{n}) is a closed string in E and W_{n} ∩ F = V_{n}. Since each W_{n} is an S-neighborhood of zero in E, it follows that V_{n} is an S-neighborhood of zero in F.

Definition 5.9. A linear map f, between two topological vector spaces E and F, is called sequentially open (S-open) if f(V) is an S-neighborhood of zero in F for each S-neighborhood V of zero in E.

Proposition 5.10. Let E, F be topological vector spaces and let f : E → F be linear, continuous and sequentially open. If E is S-barreled, then F is S-barreled.

Proof. It is clear that f is onto. Let U = (V_{n}) be a closed string in F. Then, f^{−1}(U) is a closed string in E and so f^{−1}(U) is S-topological. Since f(f^{−1}(V_{n})) = V_{n} and f is sequentially open, it follows that U is S-topological in F.

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(Received 29.11.1993)

Authors’ address:

Department of Mathematics

University if Ioannina

Ioannina, Greece